Consolidation-Expansion Operator Mechanics:A Unified Framework for Adaptive Learning
2026-05-11 • Machine Learning
Machine Learning
AI summaryⓘ
The authors propose a framework called Consolidation-Expansion Operator Mechanics (OpMech) to help learning systems decide when to stop learning new information and solidify what they know. They introduce the order-gap, a measure of how much the sequence of consolidation and expansion steps matters at a given point. This order-gap acts like a signal that shows if the system is still changing or has settled. They prove that this signal decreases as the system converges and can be used to stop learning reliably. Their framework works across different fields, including reinforcement learning and language models, offering a more principled way to decide when to stop processing compared to previous heuristics.
adaptive learning systemsconsolidation operatorexpansion operatorcommutativityorder-gapconvergencereinforcement learningstopping rulesrecursive language modelsstochastic optimization
Authors
Debashis Guha
Abstract
Every adaptive learning system must alternate between two operations: consolidating what it already knows and expanding into new evidence. We propose \emph{Consolidation-Expansion Operator Mechanics} (OpMech), a framework that makes this structure precise. The central object is the \emph{order-gap} $\Ogap(θ; e)$, the degree to which a consolidation operator~$Q$ and an expansion operator~$P_e$ fail to commute at a given knowledge state. Because the order-gap is computable from the system's own trajectory, it serves as a real-time control signal: large values indicate that the system is still sensitive to the ordering of consolidation and expansion; once the order-gap falls and stays small, further processing is unlikely to change the outcome. Three results give the signal precise meaning: the order-gap decays along convergent trajectories; a persistently large order-gap implies the system is far from its settled state; and an order-gap-based stopping rule terminates with provable guarantees in both noiseless and bounded-noise settings. The framework applies across five domains: bandits, reinforcement learning, stochastic optimization, continual learning, and recursive language models. We give conditions under which the order-gap reliably tracks convergence in three representative cases. We develop the recursive language model application in detail, showing how OpMech replaces heuristic stopping rules and fixed recursion budgets with principled, evidence-driven alternatives.